Finding the Difference of the Cubes of Two Numbers

Date: 9/29/95 at 21:57:3
From: Anonymous
Subject: Solve for x^3 - y^3
Here's the problem :
The difference of two numbers is 1.
The product of the two numbers is also 1.
What is the difference of the cubes of the numbers?

Date: 9/30/95 at 1:52:37
From: Doctor Andrew
Subject: Re: Solve for x^3 - y^3
Let x be the bigger number and y be the smaller. Then we know that
x - y = 1.
We also know that x * y = 1
So we know two things:
x - y = 1
x * y = 1
If you want to solve this algebraically, pick one of these
equations and use it to solve for either x or y in terms of the
other variable (y or x, respectively). Then substitute that
expression for x or y into the equation you didn't use.
You will get a quadratic expression (that means you'll have a term
raised to the second power) so you can use the quadratic formula to
solve it. If you're not familiar with the quadratic formula,
you can read about it here:
http://mathforum.org/library/drmath/view/53198.html
Once you've found values for x and y, you can cube them and subtract
to get the difference.
-Doctor Andrew, The Geometry Forum

Date: 9/30/95 at 14:54:47
From: Doctor Ken
Subject: Re: Solve for x^3 - y^3
Hello!
Let me offer another way you could approach this problem. I'll
give you an example of a different but similar problem:
x - y = 2
xy = 7
What is x^2 + y^2?
Well, since x-y = 2, squaring both sides gives
x^2 - 2xy + y^2 = 4.
Since xy = 7, we can add 2xy to both sides (it will get rid of the
middle term on the left, and add 14 on the right). That gives us
x^2 + y^2 = 18.
Now do you see how you could do your problem?
-Doctor Ken, The Geometry Forum